@@ -67,10 +67,12 @@ The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends weak equiv

\]

where $H_k : \ho(\Ch)\to\Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$\nbd-th homology group.

\end{paragr}

\iffalse\begin{paragr}

In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.

Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.

\end{paragr}\fi

%% \begin{paragr}

%% In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.

%% Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.

%% \end{paragr}

\begin{remark}

The adjective ``singular'' is there to avoid future confusion with another homological invariant for $\oo$\nbd-categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of $\oo$-categories} ought to be simply called \emph{homology of $\oo$\nbd-categories} as it is the only ``correct'' definition of homology. This assertation will be justified later. \todo{Le faire !}

\end{remark}

...

...

@@ -210,30 +212,30 @@ As we shall now see, when the $\oo$\nbd-categoru $C$ is \emph{free} the chain co

from the previous paragraph is an isomorphism.

\end{lemma}

\begin{proof}

\iffalse Let $G$ be an abelian group. For any $n \in\mathbb{N}$, we define an $n$-category $B^nG$ with:

\begin{itemize}

\item[-]$(B^nG)_{k}$ is a singleton set for every $k < n$,

\item[-]$(B^nG)_n = G$

\item[-] for all $x$ and $y$ in $G$ and $i<n$,

\[x \ast_i y := x +y.\]

\end{itemize}

It it straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.

%% Let $G$ be an abelian group. For any $n \in \mathbb{N}$, we define an $n$-category $B^nG$ with:

%% \begin{itemize}

%% \item[-] $(B^nG)_{k}$ is a singleton set for every $k < n$,

%% \item[-] $(B^nG)_n = G$

%% \item[-] for all $x$ and $y$ in $G$ and $i<n$,

%% \[x \ast_i y := x +y.\]

%% \end{itemize}

%% It it straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.

This defines a functor

\[

\begin{aligned}

B^n : \Ab&\to n\Cat\\

G &\mapsto B^nG,

\end{aligned}

\]

which is easily seen to be right adjoint to the functor

\[

\begin{aligned}

n\Cat&\to\Ab\\

X &\mapsto\lambda_n(X).

\end{aligned}

\]

\fi

%% This defines a functor

%% \[

%% \begin{aligned}

%% B^n : \Ab &\to n\Cat\\

%% G &\mapsto B^nG,

%% \end{aligned}

%% \]

%% which is easily seen to be right adjoint to the functor

%% \[

%% \begin{aligned}

%% n\Cat &\to \Ab\\

%% X &\mapsto \lambda_n(X).

%% \end{aligned}

%% \]

%%

Notice first that for any $\oo$\nbd-category $C$, we have $\lambda_n(\tau_{\leq n}^s(C))=\lambda_n(C)$. Suppose now that $C$ is free with basis $\Sigma=(\Sigma_n)_{n \in\mathbb{N}}$. Using Lemma \ref{lemma:adjlambdasusp} and Lemma \ref{lemma:freencattomonoid}, we obtain that for any abelian group $G$, we have

@@ -574,11 +576,11 @@ The following proposition is an immediate consequence of Theorem \ref{thm:cisins

\end{tikzcd}

\]

A throrough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Proposition \ref{prop:hmlgyderived} enables us to give the following descritption of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$

with the abelianization functor, we obtain $2$-morphism

\[

\lambda c_{\oo} N_{\oo}\Rightarrow\lambda.

...

...

@@ -706,8 +708,8 @@ Another consequence of the above counter-example is the following result, which

\]

which we know is impossible.

\end{proof}

\begin{paragr}

Even though triangle is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd-square

\begin{paragr}\label{paragr:defcancompmap}

Even though triangle \eqref{cmprisontrngle}is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd-square

Since $\J$ is nothing but the identity on objects, for any $\oo$\nbd-category $C$, the natural transformation $\pi$ yields a map

\[

\pi_C : \sH^{\sing}(C)\to\sH^{\pol}(C),

\]

which we shall refer to as the \emph{canonical comparison map.}

\end{paragr}

\begin{remark}

When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\alpha^{\Ch}$ of the morphism of $\Ch$

\[

\lambda c_{\oo}N_{\oo}(C)\to\lambda(C)

\]

induced by the co-unit of $c_{\oo}\dashv N_{\oo}$.

\end{remark}

% This motivates the following definition.

\begin{definition}

An $\oo$\nbd-category $C$ is said to be \emph{\good{}} when the canonical comparison map

...

...

@@ -754,19 +763,35 @@ Another consequence of the above counter-example is the following result, which

\]

is an isomorphism of $\ho(\Ch)$.

\end{definition}

\begin{remark}

When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\alpha^{\Ch}$ of the morphism of $\Ch$

\[

\lambda c_{\oo}N_{\oo}(C)\to\lambda(C)

\]

induced by the co-unit of $c_{\oo}\dashv N_{\oo}$.

\end{remark}

\begin{paragr}

Examples of \good{}$\oo$-categories will be presented later. The rest of this dissertation is devoted to the study of \good{}$\oo$\nbd-categories. Note that if we think of $\oo\Cat$ as a model for the homotopy theory of spaces via Thomason equivalences (which is an informal way of stating Theorem \ref{thm:gagna}), then it follows from Proposition \ref{prop:polhmlgynotinvariant} that the polygraphic homology of an $\oo$\nbd-category, up to Thomason equivalence, is not well defined. With this perspective, polygraphic homology can be thought of a way to compute singular homology of \good{}$\oo$\nbd-category.

\begin{paragr}

The rest of this dissertation is devoted to the study of \good{}$\oo$\nbd-categories. Examples of such $\oo$\nbd-categories will be presented later. Note that if we think of $\oo$\nbd-categories up to Thomason equivalences as spaces (which is an informal way of stating Theorem \ref{thm:gagna}), then it follows from Proposition \ref{prop:polhmlgynotinvariant} that polygraphic homology is not well defined. With this perspective, polygraphic homology can be thought of a way to compute singular homology of \good{}$\oo$\nbd-categories.

\end{paragr}

\section{A criterion to detect \good{}$\oo$\nbd-categories}

We shall now proceed to give an abstract criterion to find \good{}$\oo$-categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$-categories.

\begin{paragr}

Similarly to \ref{paragr:cmparisonmap}, the morphism of localizers

\[

(\oo\Cat,\W^{\folk})\to(\oo\Cat,\W^{\Th})

\]

induces a morphism of op-prederivators

\[

\J : \Ho(\oo\Cat^{\folk})\to\Ho(\oo\Cat^{\Th})

\]

such that the triangle

\[

\gamma^{\Th}=\J\circ\gamma^{\folk}

\]

is commutative. Moreover, all the constructions from \ref{paragr:defcancompmap} may be reproduced \emph{mutatis mutandis} at the level of op-prederivators. In particular, we obtain a $2$\nbd-morphism of op-prederivators